Wikipedia's description of a brachistochrone curve is
the one lying on plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time.
I was wondering if there was a counterpart to the brachistochrone curve, which would be the curve such that, given the aformentionned conditions, a bead would travel from A to B in the longest, finite amount of time.
I assume such curve would be some kind of concave version of the brachistochrone.